# Need help with a special relativity problem

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1. Jul 13, 2014

### thetrice

The example given is as follows : we have 3 space ships(numbered from left to right) placed in space moving at velocity V in the same direction (they are moving to the right) , each one can send a pulse of light to the others and the others can determine time reached. so each of them is an observer , there is a fourth observer standing on earth(which is still according to his reference).

Now we want to determine who appears to have slower time or faster according to fourth observer , so the following experiments are being done :
observer 2 (the middle space ship) sends 2 light pulses to space ship 1(first spaceship) and 2(last ship on the right) according to the first 3 observers they all appear still to each other. so they will see light reaching from 2->1 and 2->3 , at the same time X , now observer 4 sees them all 3 moving with velocity V , when the even occurred according to him, the light reached to 1 first observer before reaching the third, but since 1st observer recorded a longer time , then time runs faster to him and the third observer will record a time shorter than what should be recorded from fourth perspective so time runs slower at 3rd.

Conclusion from 1st experiment: Time runs slower at 3rd and faster at 1st.

Now repeat experiment but sending the pulse from the 1st observer to the 2nd and 3rd.
according to the first three everything is fine and running at the same speed , according to fourth observer, light will reach 2nd and third in longer time than that recorded by them meaning time is slower in 2nd and third.

Conclusion from 2nd experiment: Time runs slower at 2nd and at 3rd.

No contradictions so far Now here is the problem :

Repeating the procedure by sending pulse this time from third observer to the 1st and 2nd, again for them everything appears to run at same time , but for fourth light will reach 2nd and first in shorter time than that recorded by 1st and 2nd , so that means that time runs faster at both 1st and 2nd.

Conclusion from 3rd experiment: Time runs faster at 1st and at 2nd.

The contradiction lies between experiment 2 and 3 where time cant be faster and slower at the same time for observer 4 regarding observer 2 .

So any ideas ?

Thanks and Best Regards,
Omar Shaaban Ramzy.

2. Jul 13, 2014

### HallsofIvy

relative to the fourth observer on earth?

Every observer is "still according to his reference"! I believe you are saying that everything is to calculated relative to this fourth observer.

You have already said that they are all moving at speed V (relative to the fourth observer?) so should all have the same "time rate" according to that observer.

3. Jul 13, 2014

### thetrice

so according to the fourth observer what happens when 1st sends to the 2nd and third and when third sends to 1st and 2nd ?

4. Jul 13, 2014

### Staff: Mentor

This would be a really good time to draw a space-time diagram to see exactly what's going on. Regular poster ghwellsjr has software for drawing these, and if he's around today he'll probably chime in.

The other thing you should try is to write down the times and positions of emission and reception of each light signal using the frame in which the the spaceships are at rest, then use the Lorentz transforms to convert the coordinates of these into the frame of the fourth observer. Look carefully at these, don't forget relativity of simultaneity for all events that happen at different spatial locations, and you should be able to find the correct results:
- All observers agree that all three spaceship clocks run at the same speed.
- In the frame in which the fourth observer is at rest, the spaceship clocks are all running slow compared with the clock that is at rest in that frame.
- In the frame in which the three ships are at rest, the fourth observer's clock is running slow compared with the three spaceship clocks.

If you get different results than this... Show your work and we'll be able to help you spot where your calculation went astray.

Last edited: Jul 13, 2014
5. Jul 13, 2014

### ghwellsjr

I think your problem is that you are relying on what the earth observer actually sees and is using his own clock to establish the times that the remote events occur without taking into account the time it takes for the images to transit from those remote events to the earth observer.

As Nugatory said, I will draw some spacetime diagrams, but I won't have them ready for at least a day. In the meantime, it would be good for you to think about the light transit times and how that might resolve the discrepancies in your analysis.

6. Jul 14, 2014

### ghwellsjr

I'm going to take a stab at trying to interpret what your reasoning is and then you can tell me if I'm understanding you correctly.

Here's a spacetime diagram for the rest frame of the three spaceships which I have spaced 2000 feet apart in their rest frame. Space ship 1 is depicted in red, 2 is in black and 3 is in green. The middle one has sent a pulse of light to the two outside space ships. The light pulses arrive in 2 microseconds since the outside space ships are 2000 feet away from the middle one and the speed of light is 1000 feet per microsecond. The earth observer 4 is shown in blue moving away at -0.6c:

Here is the first spacetime diagram transformed to a speed of -0.6c so that the blue earth observer 4 is now stationary and the space ships are traveling to the right at 0.6c:

I think what you are noticing is that what took 2 microseconds in the rest frame of the space ships takes 1 microsecond for space ship 1 (red) and 4 microseconds for space ship 3 (green) and this is why you came to your conclusion, correct?

Here's the second experiment's spacetime diagram for the rest frame of the space ships:

It takes 2 microseconds for the light pulse to get from 1 (red) to 2 (black) and 4 microseconds to get from 1 (red) to 3 (green).

Here's the second experiment's spacetime diagram for the rest frame of the blue earth observer 4:

For the earth observer, the times are now 4 microseconds to get to space ship 2 (black) and 8 microseconds to get to space ship 3 (green), double what it took in the space ship's frame, therefore your conclusion that time is slower in this case, correct?

Here's the third experiment's spacetime diagram for the rest frame of the space ships:

It takes 2 microseconds for the light pulse to get from 3 (green) to 2 (black) and 4 microseconds to get from 3 (green) to 1 (red).

Finally, we have the third experiment's rest frame for the earth observer:

For the earth observer, the times are now 1 microsecond to get to space ship 2 (black) and 2 microseconds to get to space ship 1 (red), half what it took in the space ship's frame, therefore your conclusion that time is faster in this case, correct?

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7. Jul 14, 2014

### thetrice

Thanks ghwellsjr, a far well written reply , yes you adequately describe my concern.

Now this actually describes my question perfectly,My question is would the earth observer see ship number 2(black) to be faster or slower ? or that only depends on from where i send the pulse ?

as sometimes it appears to me to be slower and sometimes faster in time,but ship 1(red) and 3(green) always appear to be running with constant different time rate for observer 4(blue), while observer 2(black) has variable time rate to observer 4(blue).

Again much thanks for your effort.

8. Jul 15, 2014

### ghwellsjr

Good, I was hoping so.

It's not from where the pulse is sent but rather the direction in which it is sent. The light pulses that are directed away from the earth observer take longer to make the trip according to the earth observer's rest frame than they do according to the ships' rest frame. And the light pulses that are directed toward the earth observer take less time to make the trip according to the earth observer's rest frame than they do according to the ships' rest frame.

Since you are always comparing the light pulse transit times from one ship to another, you can't attribute the differences to just one ship. It only appears that the black ship is different because it can receive light pulses from either direction, since it is in the middle. The other two ships can only receive light pulses from one direction.

You're welcome and I hope these added comments clarify the issue for you.